m Appendix VI. 1199 In pairs as shewn, but can of course be constructed with single cylinders. The half-speed shaft K and the valves H j are of the usual type. P. 898. Balancing.—Professor Dalby has devised an ingenious graphic method for finding where additional masses, reciprocating or rotating, should be placed to complete the balance of an imperfectly balanced engine. Revolving Weights.—Referring to Fig. 10570, the crank shaft a b has a centrifugal force F, due to rotation of the crank d and the weight w. Place a drawing-board or reference plane transversely at any point a in the shaft. Then, transferring the effect of F to the plane, we have (1) A Force F (shewn dotted) acting at a (2) A Moment F^r round point a. Calculating in this manner for any number of cranks, two sets of radial values are obtained, as at G and H, one of forces and the other of moments. For purposes of comparison, w may be substituted for its centrifugal force, and two polygons are drawn, as at j and K ; (3) A Polygon of forces f^f^ &c. (4) A Polygon of moments fa^flal &c. When both polygons close, there is perfect balance of the revolving weights. In the figures the closures are incomplete. Reciprocating Weights.—If these be balanced by other recipro- cating weights, the previous construction may be exactly followed : the weights being considered as collected at the crank pins; but it is quite impossible to balance reciprocating by revolving weights. Transferring their effect to a, we have, for the reciprocating weights^ (5) A Polygon of forces. (6) A Polygon of moments. And the polygons must both close for perfect balance. To rectify imperfect Balance.—(7) Supply a force at a equal in magnitude and direction to .the closing line in the force polygon ; /.'.I «< !'